Lecture course: Metric projective

geometry

Plan: Definition and basic properties of projective structure.Application: Isometries of Hilbert metrics

Projective invariant equations. Application: Topology inthe 2-dimensional case

Local normal forms in dimension 2. Application: Solutionof the problems stated by Sophus Lie in 1882.

Projectively invariant tenrors. Application: Proof ofprojective Lichnerowicz conjecture

Conifications. Application: solution of Weyl-Ehlersproblem

Open problems, generalizations, and possible analogies inthe Finsler geometry

What is a projective structure?

Informal and inefficient definition: Projective structure is asufficiently big︸ ︷︷ ︸

to be explained

family of curves that after a reparameterisation can

be geodesics of some affine connection.

Sufficiently big: In any point in any direction there exists a curvefrom the family passing through this point in this direction.

Simplest example: the set of all straight lines on R2: it is

sufficiently big, and they are geodesics of the flat (and not only ofthe flat) connection.

Efficient definition of projective structure requires theory

Let us first study the following question: Suppose we have twosymmetric affine connections, ∇ = (Γijk) and ∇ = (Γijk). When each

geodesic of ∇, possibly after a reparameterization, is a geodesic of ∇?

Def. Two connections are ∇ and ∇ said to be projectively equivalent, ifany geodesic of ∇, possibly after a reparameterization, is a geodesic of ∇.

The above question using the new terminology: Reformulateprojective equivalence of ∇, ∇ as an easy-to-check condition

Theorem 1 (deep classics: Levi-Civita 1896, Weyl 1924). ∇ = (Γijk)

is projectively equivalent to ∇ = (Γijk), if and only if there exists an1-form φ = φi such that

Γijk = Γijk + φkδij + φjδ

ik . (∗)

The condition (∗) in the index-free form: for any vector X and anyvectorfield V ,

∇XV −∇XV = φ(X )V + φ(V )X (∗∗)

Easy control question to the audience:

Theorem 1 (deep classics: Levi-Civita 1896, Weyl 1924). ∇ = (Γijk ) is projectively equivalent to

∇ = (Γijk ), if and only if there exists an 1-form φ = φi such that

Γijk = Γ

ijk + φkδ

ij + φjδ

ik . (∗)

Question: please answer now: Does there exist two projectivelyequivalent symmetric affine connections on R

2 (or Rn) such thatthey coincide in some neighborhood and are different in anotherneighborhood?

Of course YES!!! Take a 1-form which is not zero in aneighborhood but is zero outside the neighborhood, take anyconnection ∇ = (Γijk) and “deform” it by (∗).

I will prove Theorem 1 in one direction; the seconddirection is your homework, I will give some hints

Let us first study the following natural question: What is adifferential equation of a reparameterized geodesic?

We all know that the differential equation of the geodesic parameterizedby the natural (“affine”) parameter is

∇γ γ = 0

(Or, in the index notation, γ i + Γijk γj γk = 0.)

Claim. Any reparameterised geodesic satisfies the equation∇γ γ = α(t)γ for a certain function α(t). Any regular curve satisfyingthis equation is a reparameterized geodesic.

Physical interpretation: ∇γ γ is an acceleration, so geodesics are thetrajectories of particles with no acceleration (= freely falling particles).The condition ∇γ γ = α(t)γ means that at every point the acceleration isproportional to the velocity, which implies that the particles go along thesame trajectory as in no acceleration case but the speed is not constant.

Claim. Any reparameterised geodesic satisfies the equation ∇γ γ = α(t)γ for a certain function α(t).Any regular curve satisfying this equation is a reparameterized geodesic.

Proof in direction =⇒ . We assume that the curve γ(t) is a geodesic,and that the curve γ(τ) is the geometrically the same curve but otherparameterized: ∃τ(t) such that γ(τ(t)) = γ(t).

We denote the t-derivative by dot, the τ derivative by prime, and weclearly have γ = τ γ′.

Then, the geodesic equation 0 = ∇γ γ reads

0 = ∇γ γ = ∇τγ′(τ γ′) = τ γ′ + (τ)2∇γ′ γ′.

We see that the curve γ satisfies the equation

∇γ′ γ′ = α(t)γ′ (∗ ∗ ∗)

with the function α(τ) = − τ(τ)2

.

Proof in direction ⇐=. Just observe that all steps in the proof in the

”=⇒“-direction are invertible: if we have a regular γ(τ) such that curvesuch that (∗ ∗ ∗) is satisfied, find a function τ(t) such that α = − τ

(τ)2, its

existence follows from the theory of ODE, and then go upwards along theformulas in the proof in =⇒-direction.

Proof of Theorem 1 in direction ⇐=

Theorem 1 (deep classics: Levi-Civita 1896, Weyl 1924). ∇ = (Γijk ) is projectively equivalent to

∇ = (Γijk ), if and only if there exists an 1-form φ = φi such that for all vector fields

Γijk = Γ

ijk + φkδ

ij + φjδ

ik . (∗)

We assume that ∇ = (Γijk) and ∇ = (Γijk) are related by (∗), our goal isto show that they they are projectively equivalent. That is, we need toshow that any ∇-geodesic γ, after some reparameterization, is a geodesicof ∇. Because of the Claim on the previous slide we need to show that∇γ γ = α(t)γ. We obtain this formula by direct calculation: we do it inthe index form

0 = ∇γ γ = γ+Γijk γk γj = γ+(Γijk γ

k γj)+(φkδij+φjδ

ik)γ

j γk = ∇γ γ+2φ(γ)γ

as we want.

Remark The proof may look more easier in the index-free notation: recallthat the analog of the formula (∗) is (∗∗) and is

∇XV −∇XV = φ(X )V + φ(V )X (∗∗)and using this we have

0 = ∇γ γ = ∇γ γ + φ(γ)γ + φ(γ)γ

as we want. For the proof in the =⇒-direction, which I leave you as ahomework, I recommend you though to use the index notation.

Homework. Prove theorem in the =⇒-direction.

Hint. Involves some not-completely-trivial linear algebra.

Definition of projective structure

Informal and inefficient definition: Projective structure isa sufficiently big family of curves that after a reparameteri-sation can be geodesics of some affine connection.

Efficient Def. By projective structure we understand theequivalence class of symmetric affine connections with respect tothe equivalence relation “projective equivalence”.

Remark. From Theorem 1 it follows that two symmetric affineconnection correspond to one projective structure, iff theirdifference has the form φkδ

ij + φjδ

ik for an 1-form φ.

2-dim projective structures and second order ODEs

In dimension n = 2, because of the symmetries Γijk = Γikj , the components

of Γikj in coordinates are n2(n+1)2 = 6 function. The freedom in choosing

the connection in the projective class is n = 2 “functions” (φ1, φ2). Thus,locally, projective structure is given by 4 functions of the coordinates. Letus give, following Beltrami 1859, a geometric sense to these 4 functions.

Theorem 2. Let[

Γijk

]

be a projective structure on U ⊂ R2(x , y).

Consider the following second order ODE

y ′′ = −Γ211︸ ︷︷ ︸

K0

+(Γ111 − 2Γ212)︸ ︷︷ ︸

K1

y ′ + (2Γ112 − Γ222)︸ ︷︷ ︸

K2

y ′2 + Γ122︸︷︷︸

K3

y ′3. (1)

Then, for every solution y(x) of (1) the curve (x , y(x)) is a(reparametrized) geodesic.

Corollary. The coefficients K0, ...,K3 of ODE (1) contain all theinformation of the projective structure: two connections are projectivelyrelated iff the corresponding functions K0, ...,K3 coincide.

Homework. Prove the Corollary: show that the kernel of the (linear)mapping Γijk 7→ (K0,K1,K2,K3) consists of tensors T

ijk = φkδ

ij + φjδ

ik .

Theorem 2. Let[

Γijk

]

be a projective structure on U ⊂ R2(x, y). Consider the following second order ODE

y′′

= −Γ211

︸ ︷︷ ︸

K0

+ (Γ111 − 2Γ

212)

︸ ︷︷ ︸

K1

y′+ (2Γ

112 − Γ

222)

︸ ︷︷ ︸

K2

y′2

+ Γ122

︸︷︷︸

K3

y′3. (1)

Then, for every solution y(x) of (1) the curve (x, y(x)) is a (reparametrized) geodesic.

Example. The flat projective structure [Γijk ≡ 0] corresponds to the ODEy ′′ = 0. The solutions of this ODE are y(x) = ax + b, and the curvesx 7→ (x , y(x)) = (x , ax + b) are indeed straight lines.

Remark 1. Note that the set of curves of the form (x , y(x)) is quite big:at any point in any direction there exists such a curve passing throughthis point in this direction.

Remark 2. We see a special feature of geodesics of affine connections:they are essentially the same as solutions of 2nd order ODEy ′′ = F (x , y , y ′) such that the right hand side is polynomialin y ′ of degree ≤ 3. In particular, taking an ODE y ′′ = F (x , y , y ′)such that F is not a polynomial in y ′ of degree ≤ 3, the set of thecurves of the form (x , y(x)) are geodesics of no affine connection. Wewill return to this problem in the last lecture.

How many geodesics determine the projective structure?

We will answer this question in dimension 2, and give an application inthe next slides.We consider a projective structure

[

Γijk

]

and the corresponding ODE

y ′′ = −Γ211︸ ︷︷ ︸

K0

+(Γ111 − 2Γ212)︸ ︷︷ ︸

K1

y ′+(2Γ112 − Γ222)︸ ︷︷ ︸

K2

y ′2+ Γ122︸︷︷︸

K3

y ′3. (1)

Claim. For any point (x , y), 4 different geodesics passing through thispoint determine the coefficients K0(x , y),K1(x , y),K2(x , y),K3(x , y) atthis point.

Proof. 4 different geodesics passing through (x , y) correspond to 4different solutions y1, y2, y3, y4 of (1) such that yi (x) = y . Knowinggeodesics implies that we know y ′

i (x) and y ′′i (x) which implies that we

known the values of the polynomial

P(y ′) = K0(x , y) + K1(x , y)y′ + K2(x , y)y

′2 + K3(x , y)y′3

at four points y ′1(x), ..., y

′4(x) and we know that values at 4 points

determines a polynomial of degree ≤ 3.

Application in Finsler geometry: proof of the 2-dim de laHarpe conjecture

Let K ⊂ Rn be a compact convex body.

Hilbert metric is the following distance func-tion d : int(K )× int(K )→ R on the interiorof K : for x 6= y ∈ int(K ) we consider thestraight line containing x , y and denote by xand y the intersection points of this straightline with the boundary of K .

Then, we put

d(x , y) := ln((x , x ; y , y)

):= ln

( |y − x ||y − y | :

|x − x ||x − y |

)

,

where | · | denotes the usual Euclidean length.

Known properties of the Hilbert metrics

Hilbert metric is a Finsler metric, the corre-sponding Finsler function is given by

F (x , v) = |v ||x−x+| +

|v ||x−x−| .

Straight line segments are geodesics

Projective transformations preserving the convex body preserve theHilbert metric

Remark. If the boundary is not strictly convex, the geodesics are notnecessary unique.

Question of de la Harpe (1991)

For what K all isometries of K are projective transformations?

Example (de la Harpe): If K is simplex, there exist isometries thatdo not come from projective transformations.

If K is strictly convex, any isometry is a projective transformation(deep classics; possibly Hilbert)

2011: Answer by Walsh and Lemmens for polyhedral K .

2013: Answer by Walsh for all convex bodies

Theorem (M∼ – Troyanov 2014, in arXive today). In dimensiontwo, each isometry φ : K → K is a projective transformation unless K isa triangle.

We see that our theorem is not new; but the proof of Walsh is relativelycomplicated and you will see that our proof is trivial for those who heardthe first part on today’s lecture.

Proof .

Fact (possibly Hilbert 1895, de la Harpe 1991). A straight linecontaining an extremal point is UNIQUE MINIMIZING geodesic.

We consider four extremal points A,B ,C ,D of K .

For every point P ∈ int(K ), we consider the in-tersections of the straight lines containing A and P(resp., B and P , C and P , D and P).

A B

C

D

P

As we learned today, these four straight lines defineuniquely a projective structure; this projective structureis projectively flat (all geodesics are straight lines)

The push-forward of this projective structure is a projective structure,since the 4 geodesics are unique, isometry maps sends them to straightlines and therefore the push-forward of the projective structure isprojectively flat and φ is a projective transformation

Lecture two: Projectively invariant differential operators

Plan

1. Definition

2. Two main examples: (projective) Killing and metrizationequations

3. Philosophy of metric projective geometry

4. Application: what 2 dim manifolds admit projectively relatedmetrics.

Projectively invariant operators: def and trivial examples

Projectively invariant = does not depend on the choice of a connection inthe projective class and on the coordinate system.

Not an Example. Covariant differentiation of vectors or tensors is NOTprojectively invariant: If we replace ∇ by a projectively equivalent ∇,then the covariant derivative will be CHANGED:

∇XV −∇XV = φ(X )V + φ(V )X .

Trivial Example. The outer derivative ω 7→ dω on the space of k-formsis projectively invariant. Indeed, it does not depend on a connection at

all. (Say, for 1-forms, d(adx + bdy) =(

∂b∂x− ∂a

∂y

)

dx ∧ dy)

Our next goal is to construct two ‘nontrivial’ projectively invariantdifferential operations; they will play an important role later todayand in other lectures but the price we need to pay now is that weneed to introduces weighted tensor fields

What are weighted tensors? What is weight?

We assume that our manifold M is orientable and fix anorientation. We consider the bundle ΛnM of positive volume formson M

Recall. Volume form is a scew-symmetic form of maximal order,Vol = f (x)dx1 ∧ ... ∧ dxn with f 6= 0. “Positive” means that if thebasis ∂

∂x1, ..., ∂

∂xnis positively oriented then f (x) > 0.

Positive volume bundle is a locally trivial 1-dimensional bundle over ourmanifold M with the structure group (R>0, ·). That means in particularthat for small neighborhood U ⊂ M we have an isomorphisms betweenΛnU and R>0 × U: there are two natural ways to choose theisomorphism, let us discuss them.

1. Choose a section in this bundle, i.e., a volume form, the othersections of this bundle can be thought to be positive functions onthe manifold (and if we change coordinates theytransform like functions, i.e., do not transform atall). This situation will be actively used later, when the volumeform is parallel with respect to an affine connection in the projectiveclass.

2. In local coordinates x = (x1, ..., xn), we can choose the volume formdx1 ∧ ... ∧ dxn, the volume form Ω = f (x)dx1 ∧ ... ∧ dxn

corresponds to the function f (x). Its transformation rule is differentfrom that of functions: a coordinate change, x = x(y) transforms

f (x) to det(

dxdy

)

f (x(y)).

(Λn)αM

Let α ∈ R \ 0. Since t → tα is an isomorphism of (R>0, ·), forany 1-dimensional (R>0, ·)-bundle its power α is well-defined and isalso an one-dimensional bundle. We consider (Λn)

αM. It is an1-dimensional bundle, so its sections locally can be viewed asfunctions. Again we have two ways to view the sections asfunctions:

1. Choose a volume form Ω, and the corresponding sectionω = (Ω)α of (Λn)

αM. Then, the other sections of this bundlecan be thought to be positive functions on the manifold.

2. In local coordinates x = (x1, ..., xn), we can choose the section(dx1 ∧ ...∧ dxn)α, then the section ω = (f (x)dx1 ∧ ...∧ dxn)αcorresponds to the function (f (x))α. Its transformation rule isdifferent from that of functions: a coordinate change,

x = x(y) transforms (f (x))α to(

det(dxdy

))α

f (x(y))α.

Weighted tensors

Def. By a (p,q)-tensor field of projective weight k we understand asection of the following bundle:

T (p,q) ⊗ (Λn)k

n+1 M (notation := T (p,q)M(k))

If we have a preferred volume form on the manifold, the sections ofT (p,q)M(k) can be identified with (p,q)-tensors fields. The identificationdepends of course on the choice of the volume form.

If we do not have a preferred volume form on the manifold, in a localcoordinate system one can choose (dx1 ∧ ... ∧ dxn) as the preferredvolume, and still think that sections are “almost” (p,q)-tensors: they arealso given by np+q functions but their transformation rule is slightlydifferent from that for tensors: in addition to the usual transformation

rule for tensors one needs to multiply by(

det(

dxdy

))α

with α = kn+1 .

Covariant differentiation of weighted tensor bundles

Fact (e.g. brute force calculations). Suppose (projectively equivalent)connections ∇ = (Γijk) and ∇ = (Γijk) are related by the formula

∇XV −∇XV = φ(X )V + φ(V )X (∗∗).

Then, the covariant derivatives of a volume form Ω ∈ Γ (ΛnM) in theconnections ∇ and ∇ are related by

∇XΩ = ∇XΩ− (n + 1)φ(X )Ω.

In particular, the covariant derivatives of the section

ω :=

(

Ωk

n+1

)

∈ Γ((Λn)k

n+1 M) are related by

∇Xω = ∇Xω − kφ(X )ω. (2)

First example of projectively invariant differential operation

∇Xω = ∇Xω − kφ(X )ω. (2)

Let K ∈ Γ(T (0,1)M(−2)

)be an 1-form of projective weight (−2).

We calculate the difference their ∇- and ∇- derivatives assuming

∇XV −∇XV = φ(X )V + φ(V )X (∗∗) :

∇XK = ∇XK −φ(X )K − K (X )φ︸ ︷︷ ︸

because of (∗∗)

+ 2φ(X )K︸ ︷︷ ︸

because of (2)

= ∇XK+φ(X )K−K (X )φ. (3)

Theorem. For (0, 1)-tensors of projective weight (-2) the operation

K 7→ Symmetrization Of(∇K ) (K1)

is projectively invariant: it does not depend on the choice of the affineconnection in the projective class.

Proof. Observe in (3) that the difference between (∇XK )(Y ) and(∇XK )(Y ) is scewsymmetric in X ,Y and vanishes after symmetrization.

Remark. In the index notation, the mapping (K1) reads

Ki 7→ Ki,j + Kj,i . The equation (K1) = 0 is called projective Killing

equation for weighted 1-forms.

Theorem. For (0, 1)-tensors of projective weight (-2) the operation

K 7→ Symmetrization Of(∇K)

is projectively invariant: it does not depend on the choice of the affine connection in the projective class.

Corollary 1. For (0, 2)-tensors of projective weight (−4) the operation

K 7→ Symmetrization Of(∇K )

is projectively invariant: it does not depend on the choice of the affineconnection in the projective class.

Proof. Decompose (0, 2) tensors of weight −4 into the sum ofsymmetric tensor product of (0, 1) tensors of weight −2 and applyCorollary 1.

Notation. The equation in Corollary 1 is called projective Killingequation; it will play important role at the end of the lecture.

One more important projectively invariant operation

Theorem. For (1, 0)-tensors of projective weight 1 the operation

σ 7→ Trace Free Part Of∇(σ) = σi,j − 1

nσs

,sδij .

is projectively invariant: it does not depend on the choice of the affineconnection in the projective class.

Proof. By calculations which are essentially the same as in Corollary 1.

Corollary 3. For symmetric (2, 0)-tensors of projective weight 2 theoperation

σij 7→ σij,k − 1

n+1 (σis,sδ

jk + σjs

,sδik) (4)

is projectively invariant: it does not depend on the choice of the affineconnection in the projective class.

Proof. Decompose (2, 0) tensors of weight 2 into the sum of symmetrictensor products of (0, 1) tensors of weight −2 and apply Corollary 2

Remark. In the index-free notation the operation (4) reads

σ 7→ Trace Free Part Of (∇σ) ,though ∇σ is a (2,1)-(weighted)-tensor and trace is a (weighted) vector.

Geometric importance of the operatorσ 7→ Trace Free Part Of (∇σ) .

(Metrization) Theorem 3 (Eastwood-M∼ 2006). Suppose theLevi-Civita connection of a metric g lies in a projective class [∇]. Then,σij := g ij ⊗ (Volg )

2n+1 is a solution of

Trace Free Part Of (∇σ) = 0. (5)

Moreover, for every solution of the equation (5) such that det(σ) 6= 0there exists a metric whose Levi-Civita connection lies in the projectiveclass.

Proof in the direction ⇒. We assume that ∇g ∈ [∇]. Since ourequation is projectively invariant, we may assume that we work in theconnection ∇g . In this connection the metric and therefore all objectsconstructed by the metric are parallel so ∇g (σ) = 0 which of courseimplies (5)

(Metrization) Theorem 3. Suppose the Levi-Civita connection of a metric g lies in a projective class

[∇]. Then, σij := g ij ⊗(Volg

) 2n+1 is a solution of

Trace Free Part Of (∇σ) = 0. (5)

Moreover, for every solution of the equation (5) such that det(σ) 6= 0 there exists a metric whoseLevi-Civita connection lies in the projective class.

Proof in the direction ⇐ is your homework: Observe that thoughequation

σij 7→ σij,k − 1

n+1

(

σis,sδ

jk + σjs

,sδik

)

does not depend on the choice of a connection in the projective class, thepart of it marked by blue color does depend. Find out how it depends andprove that there exists a connection in the projective class such that theblue part is zero, show then that this connections preserves a metric suchthat σ is obtained by the metric by the formula in Theorem 3.

Relation between (nondegenerate) solutions σ ofmetrization equations and metrics in coordinates

Let us work in a coordinate system and choose dx1 ∧ ...∧ dxn. as avolume form.

If we have a metric gij , then the corresponding solution of themetrization equation is given by

σij :=

(

g ij ⊗ (Volg )2

n+1

)

= g ij | det g |1

n+1 .

For a solution σ = σij of the metrisation equation such that itsdeterminant in not zero, the corresponding metric is given by

g ij := | det(σ)|σij .

Metrization equations in dimension 2 in coordinates:

Theorem 3 (Metrization equations in all dimensions:) Trace Free Part Of (∇σ) = 0. (5)

As we remember from Lecture 1, in dimension 2 the four functionsK0,K1,K2,K3 (which are the coefficients of the equation) (essentially R.Liouville 1889)

y′′

= −Γ211

︸ ︷︷ ︸

K0

+ (Γ111 − 2Γ

212)

︸ ︷︷ ︸

K1

y′+ (2Γ

112 − Γ

222)

︸ ︷︷ ︸

K2

y′2

+ Γ122

︸︷︷︸

K3

y′3.

encode the projective class of the connection Γijk .

In this setting, the metrization equations in the following system of 4PDE on three unknown functions:

σ22x − 2

3 K1 σ22 − 2K0 σ

12 = 0σ22

y − 2σ12x − 4

3 K2 σ22 − 2

3 K1 σ12 + 2K0 σ

11 = 0−2σ12

y + σ11x − 2K3 σ

22 + 23 K2 σ

12 + 43 K1 σ

11 = 0σ11

y + 2K3 σ12 + 2

3 K2 σ11 = 0

Corollary. Generic projective structure is not metrizable.Explanation (formal proof in Bryant et al 2009). The system isoverdertermined: 4 equations on three unknown functions, and genericoverdetermined systems have no solution.

Metric Projective Geometry: philosophy and goals

One can of course study projective structures without thinking aboutwhether there is a (Levi-Civita connection of a) metric in the projectiveclass.

Luck of “easy to formulate, hard to prove” results.

Virtually no applications in physics

Let us study metrizable projective structures, i.e., suchthat there exists a metric in the projective class.

Generic metrizable projective structure has only one, up to a scaling,metric in the projective class. In this case, all geometric questions can bereformulated as questions on this metric.

We will study metrisable projective structures suchthat there exists at least two nonproportional metrics inthe projective class.

Many “easy to formulate, hard to prove” results. Many namedproblems. Applications in physics

Many questions and methods can be generalized for Finslermanifolds

Next goal: easy to formulate result for today

Theorem (M∼-Topalov 1998). Let (M2, g) be a two-dimensionalclosed (compact, no boundary) Riemannian manifold. Assume a metric gis projectively related to g and is nonproportional to g . Then, M2 hasnonnegative Euler characteristic.

I will give an easy proof of this theorem using what we learned today.

I use: projective Killing equation is projectively invariant.

Corollary 2. For (0, 2)-tensors of projective weight −4 the operation

K 7→ Symmetrization Of(∇K)

is projective invariant.

Example. The (Levi-Civita connection of the) metric g does haveone nontrivial solution of this equation, namely

K = g ⊗ (Volg )−4n+1 .

But the projective Killing equation does not depend on the choiceof connection in the projective class.

Thus, any metric in the projective class allows usto construct a solution of the projective Killingequation. Say, if we have another metric g in thesame projective class, then

K = g ⊗ (Volg )−4n+1 .

is (still) a solution.

Geometric sense of Killing equations: conservativequantities.

Theorem 4. Suppose K is a solutions of the projective Killing equation.Then, for any metric g in the projective class the tensor field

K := K ⊗ (Volg )4

n+1 .

is a Killing tensor, this means that for any parameterized g -geodesic γthe function

t 7→ I (γ(t), γ(t)) = K (γ(t), γ(t))

is constant.Proof. We need to show that

∇γ (K (γ, γ)) = 0. (⋆)

Because of the definition of geodesic, ∇γ γ = 0, and (⋆) reduces to

∇K (γ, γ, γ) = 0,

which follows from Symmetrization Of(∇K ) = 0.

Trivial conservative quantity: energy

Example above. The following section of T (0,2)M(−4)

K = g ⊗ (Volg )−4n+1

is a solution of the projective Killing equation for g .

Theorem 4. K := K ⊗ (Volg )4

n+1 . is a Killing tensor.

If we take the K from the first frame, and use it to constract K from thesecond frame, then we obtain K = g , which is of course a Killing tensor;the corresponding conservative quantity is the kinetic energy.

Nontrivial conservative quantity, if we have twononproportional metric in the projective class

Example above. For a metric g in the projective class the following section of T (0,2)M(−4)

K = g ⊗ (Volg )−4n+1

is a solution of the projective Killing equation.

Theorem 4. K := K ⊗ (Volg )4

n+1 . is a Killing tensor.

If we take the K from the first frame, and use it to constract K from the

second frame, then we obtain K =∣∣∣det gdet g

∣∣∣

2n+1

g , which is now

nonproportional to the trivial (=always existing) Killing tensor gij . Thecorresponding is given by

I (x , ξ) =∣∣∣det gdet g

∣∣∣

2n+1

g(ξ, ξ)

Historical remark. There are of course direct proofs that I is aconservative quantity, the most classical is possibly due to Painleve 18... Iwill possibly show you other proofs in lecture VI, when I speak aboutFinlser metrics.

Proof of announced theorem

Theorem (M∼-Topalov 1998). Let (M2, g) be a two-dimensional closed (compact, no boundary) Rie-

mannian manifold. Assume a metric g is projectively related to g and is nonproportional to g . Then, M2

has nonnegative Euler characteristic.

In dimension 2, the conservative quantity constructed

I0(ξ) :=

∣∣∣∣

det(g)

det(g)

∣∣∣∣

23

g(ξ, ξ).

Assume the surface is neither torus nor the sphere. The goal is to showthat g and g are proportional.

Because of topology, there exists x0 such that g|x0 = const ·g|x0 . W.l.o.g.

we assume const = 1. We assume g|x1 6= g|x1 and find a contradiction.

Homework

Use Killing tensors to show that the space of solutions of themetrization equation is finite-dimensional (for n = 2 first if itmakes your life easier and then for any dimension since there areno two-dimensional phenomenas in the proof)

Lecture 3

Plan

Local normal forms of projectively related Riemannian metrics

Problems of Lie and their solution

How we solved the problems of Lie

Our first goal is to prove the Dini’s Theorem 1869

Local normal form question (Beltrami 1865): Given two projectivelyrelated metric, how do they look in “the best” coordinate system (near ageneric point)? How unique is such best coordinate system?

Theorem (Dini 1869). Let g and g are projectively related 2 dimRiemannian metrics. Then, in a neighborhood of almost every point thereexists a coordinate system such that in this coordinate system the metricsare

g =

(X (x)− Y (y)

X (x)− Y (y)

)

g =

(X (x)−Y (y)X (x)Y (y)2

X (x)−Y (y)X (x)2Y (y)

)

The coordinates are unique modulo (x , y) 7→ (±x + b,±y + d).

Remark. The answer in higher dimensions is also known (Levi-Civita)In other signatures the answer to the Beltrami questions is also known(Darboux/Lie for dim 2, Bolsinov-Matveev 2013 for all dimensions).

Rem. In the 2 dim case of splitted signature, there are two more cases:when g−1g has complex eiganvalues, and when g−1g has Jordan block.

Proof: coordinates such that g and g are diagonal

Such coordinates exist near every generic points:

Indeed, at the points where g is not proportional to g the (1,1)-tensorg−1g = g is gjs has two different eigenvalues. We consider the coordinatesystem (x , y) such that ∂

∂xand ∂

∂yare eigenvectors.

Since the eigenvectors are orthogonal w.r.t. g and w.r.t. g , in thiscoordinates the metrics are diagonal.

Plugging diagonal σ, σ in the metrization theorem

Thus, we may assume that in the coordinate system (x , y) the metricsare diagonal and theirfore the corresponding solutions of the metrization

equation σ =

(

g ij ⊗ (Volg )2

n+1

)

= g ij(det g)1

n+1 ,

σ =

(

g ij ⊗ (Volg )2

n+1

)

= g ij(det g)1

n+1 are diagonal.

Consider the (1,1)-tensor field A = σ(σ)−1 = σisσjs , it is also diagonal:

σ =

(σ11

σ22

)

, A =

(A1

A2

)

, σ =

(A1σ

11

A2σ22

)

.

Let us now plug these σ and σ in the metrization theorem whosetwo-dimensional version is in Lecture 2:

σ22x − 2

3K1 σ22 − 2K0 σ12 = 0

σ22y − 2σ12

x − 43K2 σ22 − 2

3K1 σ12 + 2K0 σ11 = 0

−2σ12y + σ11

x − 2K3 σ22 + 23K2 σ12 + 4

3K1 σ11 = 0

σ11y + 2K3 σ12 + 2

3K2 σ11 = 0

8 equations on 6 unknown is too much – elementary trickssolve the system

σ22x − 2

3K1 σ22 = 0

σ22y − 4

3K2 σ22 + 2K0 σ11 = 0

σ11x − 2K3 σ22 + 4

3K1 σ11 = 0

σ11y + 2

3K2 σ11 = 0

∣∣∣∣∣∣∣∣∣

A2σ22

x + (A2)xσ22 − 2

3K1 A2σ

22 = 0

A2σ22

y + (A2)yσ22 − 4

3K2 A2σ

22 + 2K0 A1σ11 = 0

A1σ11

x + (A1)xσ11 − 2K3 A2σ

22 + 43K1 A1σ

11 = 0

A1σ11

y + (A1)yσ22 + 2

3K2 A1σ

11 = 0

Message: systems of PDE with more equations are as a rule easierto solve than that with less equations

How we proceed: Solve the first 4 questions with respect to K0, ...,K3

and substitute the result in the last 4 equations. One obtains theequations

(A1)y = 0(A2)x = 0

((A1 − A2)σ11(σ22)2)x = 0

((A1 − A2)σ22(σ11)2)y = 0.

implying

A1 = X (x)A2 = Y (y)

(X (x) − Y (y))σ11(σ22)2 = 1Y1(y)

(X (x) − Y (y))σ22(σ11)2 = 1X1(x)

.

A1 = X (x)A2 = Y (y)

(X (x) − Y (y))σ11(σ22)2 = 1Y1(y)

(X (x) − Y (y))σ22(σ11)2 = 1X1(x)

.

Observe now, because of the relation g ij = | det(σ)|σ and because of thematrices σ, σ are diagonal, we have σ11(σ22)2 = g22 andσ22(σ11)2 = g11. Thus, we obtain that

g = (X − Y )(X1dx2 + Y1dy

2) and A = diag(X ,Y ).

By a coordinate change x = x(xnew ), y = y(ynew ), one can “hide” X1

and Y1 in dx2 and dy2 and obtain the formulas of Dini

Projective transformations

Def. Projective transformation of a projective structure [Γ] is a (local)diffeomorphism that preserves [Γ].

Geometric (equivalent) definition. Projective transformations arediffeomorphisms that send geodesics to geodesics.

Example. Affine transformations from 1st year linear algebra course(i.e., x 7→ Ax + b with nondegenerate matrix A) are projectivetransformations of the flat (i.e., when Γijk ≡ 0) projective structure.

Example. Projective transformations from linear algebra are projectivetransformations of the flat projective structure.

Def. A vector field is projective w.r.t. [Γ], if its (local) flow acts byprojective transformations.

Example. A Killing vector field of a metric is projective w.r.t. theprojective structure of the metric.

Beltrami example: projective algebra of the round sphereis sl(n + 1).

We consider the standard Sn ⊂ Rn+1 with the induced metric.

Fact. Geodesics of the sphere are thegreat circles, that are the intersec-tions of the 2-planes containing thecenter of the sphere with the sphere.

Beltrami (1865) observed:

For every A ∈ SL(n + 1)we construct−−−−−−−−→ a : Sn → Sn, a(x) := A(x)

|A(x)|

a is a diffeomorphism

a takes great circles (geodesics) to great circles (geodesics)

a is an isometry iff A ∈ O(n + 1).

Thus, Sl(n + 1) acts by projective transformations on Sn. Its stabilizatoris discrete and therefore the algebra of projective vector fields issl(n + 1); in dimension n = 2 it has dimension (n + 1)2 − 1 = 8.

Example of Lagrange 1789

0

f(X)X

Radial projection f : S2 → R2

takes geodesics of the sphere togeodesics of the plane, becausegeodesics on sphere/plane are in-tersection of plains containing 0with the sphere/plane.

Thus, the projective structure of the plane is the same as that of thesphere and also has 8-dimensional projective algebra.Everything survives to all dimensions and all signatures and for negativecurvature

“Nice” result of todays lecture: Problems of Lie

Lie 1882:Problem I: Es wird verlangt, die Form des Bogenelementes einerjeden Flache zu bestimmen, deren geodatische Kurven eine infini-tesimale Transformation gestatten.

English translation:

Describe all 2 dim metrics admitting

Problem I: one projective vector field

Problem II: many projective vector fields

Lie 1882:Problem II: Man soll die Form des Bogenelementes einer jedenFlache bestimmen, deren geodatische Kurven mehrere infinitesi-male Transformationen gestatten.

Both problems are local, in a neighborhood of a generic point

Solution of the 2nd Lie Problem

Theorem (Bryant, Manno, M∼ 2007) If a two-dimensional metric gof nonconstant curvature has at least 2 projective vector fields such thatthey are linear independent at the point p, then there exist coordinatesx , y in a neighborhood of p such that the metrics are as follows.

1. ε1e(b+2) xdx2 + ε2be

b xdy2, whereb ∈ R \ −2, 0, 1 and εi ∈ −1, 1

2. a(

ε1e(b+2) xdx2

(eb x+ε2)2+ eb xdy2

eb x+ε2

)

, where a ∈ R \ 0,b ∈ R \ −2, 0, 1 and εi ∈ −1, 1

3. a(

e2 xdx2

x2 + ε dy2

x

)

, where a ∈ R \ 0, and ε ∈ −1, 1

4. ε1e3xdx2 + ε2e

xdy2, where εi ∈ −1, 1,

5. a(

e3xdx2

(ex+ε2)2+ ε1e

xdy2

(ex+ε2)

)

, where a ∈ R \ 0, εi ∈ −1, 1,

6. a(

dx2

(cx+2x2+ε2)2x+ ε1

xdy2

cx+2x2+ε2

)

, where a > 0, εi ∈ −1, 1, c ∈ R.

Theorem (M∼ 2008): Let v be a projective vector field on (M2, g).Assume the restriction of g to no neighborhood has an infinitesimalhomothety. Then, there exists a coordinate system in a neighborhood ofalmost every point such that certain metric g geodesically equivalent tog is given by

1. ds2g = (X (x)− Y (y))(X1(x)dx2 + Y1(y)dy

2), v = ∂∂x

+ ∂∂y

, where

1.1 X (x) = 1x, Y (y) = 1

y, X1(x) = C1 · e

−3x

x, Y1(y) =

e−3y

y.

1.2 X (x) = tan(x), Y (y) = tan(y), X1(x) = C1 · e−3λx

cos(x) ,

Y1(y) =e−3λy

cos(y) .

1.3 X (x) = C1 · eνx , Y (y) = eνy , X1(x) = e2x , Y1(y) = ±e2y .2. ds2g = (Y (y) + x)dxdy , v = v1(x , y)

∂∂x

+ v2(y)∂∂y

, where

2.1 Y = e32y ·

√y

y−3 +∫ y

y0e

32ξ ·

√ξ

(ξ−3)2 dξ,

v1 =y−32

(

x +∫ y

y0e

32ξ ·

√ξ

(ξ−3)2 dξ)

, v2 = y2.

2.2 Y = e−32λ arctan(y) ·

4√

y2+1

y−3λ +∫ y

y0e−

32λ arctan(ξ) ·

4√

ξ2+1

(ξ−3λ)2 dξ,

v1 =y−3λ

2

(

x +∫ y

y0e−

32λ arctan(ξ) ·

4√

ξ2+1

(ξ−3λ)2 dξ

)

, v2 = y2 + 1.

2.3 Y (y) = yν , v1(x , y) = νx , v2 = y .

Why the problems of Lie were not solved before? Whatknow-how allowed us to solve it?

Many people tried (including Lie and his students)

One immediately reformulates the problem as a (quasilinear)2ND ORDER system of PDE on the components of metricand of the vector field; the system is to hard to solve by hands.

Our new viewpoint on the problem which allowedto solve it was to use projectively-invariantobjects.

This allowed to reduced the PDE-reformulation to MOREequations of the 1ST order which can be solved by hands.

How we solve the problems of Lie: first observation:

We had two projectively invariant equations: Killing equations andmetrization equations: let us compare them in dimension 2:

Metrization equation in dimension 2:

σ22x − 2

3K1 σ22 − 2K0 σ12 = 0

σ22y − 2σ12

x − 43K2 σ22 − 2

3K1 σ12 + 2K0 σ11 = 0

−2σ12y + σ11

x − 2K3 σ22 + 23K2 σ12 + 4

3K1 σ11 = 0

σ11y + 2K3 σ12 + 2

3K2 σ11 = 0

Killing equation in dimension 2:

a11x − 23K1 a11 + 2K0 a12 = 0

a11y + 2 a12x − 43K2 a11 + 2

3K1 a12 + 2K0 a22 = 0

2 a12y + a22x − 2K3 a11 − 23K2 a12 + 4

3K1 a22 = 0

a22y − 2K3 a12 + 23K2 a22 = 0

We see that the equations coincide after renaming the variables:

(a11 a12a12 a22

)

= Comatrix

((σ11 σ12

σ12 σ22

))

=

(σ22 −σ12

−σ12 σ11

)

. (∗)

Remark. The operation of taking the comatrix is a “geometric”operation: it does not depend on the coordinate system, it is invertible,and in dimension 2 it gives a linear bijection between (2, 0)-tensors ofprojective weight 2 and (0, 2)-sections of projective weight −4.

We just have proved the following theorem:

Theorem. In dimension 2, solutions of metrization equations are inone-to-one correspondence to the solutions of projective Killing equations(Killing tensors are assumed symmetric), the correspondence is given incoordinates by

(a11 a12a12 a22

)

= Comatrix

((σ11 σ12

σ12 σ22

))

. (∗)

Corollary. Suppose there exists a solution σij of the metrisationequation such that aij is degenerate but nonzero (in every point). Then,there exists a (projective) Killing 1-form, and in the case we have ametric in the projective class, a Killing vector field.

Proof. In dimension 2, comatrix of a nonzero degenerate matrix is anozero degenerate matrix. It has therefore rank 1 and degenerate nonzeroprojective Killing tensor has the form a = ±K ⊗ K for some 1-form K .This 1-form is Killing: one can see it from equations but let us see itgeometrically in metric situation which is sufficient for our goals.

The corresponding conservative quantity is K (ξ) =√

±a(ξ, ξ). Since thefunction a(ξ, ξ) is preserved along geodesics, the function K (ξ) is alsopreserved along geodesics, so that K is a Killing one form (and afterraising the index we obtain a Killing vector field).

We may assume that our metrics do not have Killingvector fields

Corollary. Suppose there exists a solution σij of the metrisation equation such that aij is degeneratebut nonzero (in every point). Then, there exists a (projective) Killing 1-form, and in the case we have ametric in the projective class, a Killing vector field.

Killing vector field is automatically projective, so withoutloss of generality in the solution of the first Lie problem weassume nonexistence of a Killing vector field and thereforewe assume that all nonzero solutions of the metrizationequation we meet are nondegenerate.

Lie derivative w.r.t. projective vector field sends solutionsof the metrization equations to solutions

Claim. Let v be projective for [Γ] and suppose σ is a solution of themetrization equation. Then, Lvσ is also a solution of the metrizationequation.

Proof in the 2 dim case. Everything is coordinate-invariant, so w.l.o.g.we can work in coordinates (x , y) such that v = ∂

∂x. In this coordinates,

the coefficients K0, ...,K3 do not depend on x , so the coefficients of themetrization equation

σ22x − 2

3K1 σ22 − 2K0 σ12 = 0

σ22y − 2σ12

x − 43K2 σ22 − 2

3K1 σ12 + 2K0 σ11 = 0

−2σ12y + σ11

x − 2K3 σ22 + 23K2 σ12 + 4

3K1 σ11 = 0

σ11y + 2K3 σ12 + 2

3K2 σ11 = 0

do not depend on x as well. Then, for any solution σij the∂∂x-Lie derivative (will be explained on trivial language on

the next slide), which is simply ∂∂x

is also a solution

Remark. The proof actually works in other dimensions as well – wesimply need to observe that in coordinates (x1, ..., xn) such that v = ∂

∂x1

the coefficients of the metrization equation does not depend on x1.

The action of the Lie derivative of a projective vector fieldon the space of solutions of the metrization equation

Notation. We denote the space of solutions of metrization equation bySol([Γ]).Fact (Liouville 1889 in dim 2, Sinjukov 1959 in dim n), will bepossibly explained later and was your homework:

dim (Sol([Γ])) ≤ (n+1)(n+2)2 <∞.

Consider the linear mapping Lv : Sol([Γ])→ Sol([Γ]). Well-definedbecause by Claim above Lie derivative of a solution is a solution.

Fact from linear algebra. If dim (Sol([Γ])) ≥ 2, there exists atwo-dimensional invariant subspace of Lv , we will work with this subspaceand forget the rest.

By linear algebra, there exists a basis σ, σ ∈ Sol([Γ]) such that in thisbasis the matrix of Lv is given by the following (real) Jordan normal form.

(λ1

λ2

)

,

(λ 1

λ

)

,

(α β−β α

)

.

As we explained above, w.l.o.g. we may assume that σ and σ arenondegenerate; then they correspond to some metrics.

Assume that the matrix of Lv is

(λ1

λ2

)

, and assume in addition that

σ, σ ∈ (Sol([Γ])) corresponds to Riemannian metrics. Then, by the DiniTheorem the metrics g and g corresponding to σ and σ are given by

g =

(X (x)− Y (y)

X (x)− Y (y)

)

g =

(X (x)−Y (y)X (x)Y (y)2

X (x)−Y (y)X (x)2Y (y)

)

Observe that the projective vector field v preserves the pencil of thesolutions ασ + βσ, and therefore any object constructed by thesesolutions, in particular the lines of the coordinates (x , y). Then, thevector field v = (v1(x), v2(y)). Now, from Lvσ = λσ it follows that vpreserves the conformal structure of the metric g , so it is a holomorphicvector field. Then, it is constant or linear, i.e., up to a coordinate changeand factor it is either

v = (x , y) or v = (const1, const2).

In both cases the flow is given by precise formulas, which after some workgive formulas for X (x) and Y (y) from case 1.1. of Theorem.

Few words about the solution of the second problem of Lie

Recall: Difference between the first and the second problem: in the 1stproblem we look for metrics with one projective vector field, and in the2nd problem with many.

One should of course check whether the metrics we obtained do nothave another projective vector field.

This is an algorithmically doable problem assuming we candifferentiate and arithmetic operations.

The algorithm is essentially due to Lie and is build in Mapleand since the metrics in Theorem are explicit Maple can workwith them and gives an answer.

Then, the only additional problem to solve is to omit theassumption that their exists no Killing vector field. But if thereexists a Killing vector field, one can use again projective invarianceof the Killing equation (I do not go into details at this point)

Historical Remark. In this lecture I first solved the 1st problem, and thenused it in the 2nd problems of Lie, historically first the 2nd problem was solved(Bryant, Manno, M∼ 2006) and then the 1st (M∼ 2008); but the solution ofthe 2nd without having the 1st is computationally quite hard.

What Lie did not know? Why he did not solved hisproblems himself?

Ingredients of the proof.

Local normal forms of projectively equivalent metrics? Lie knew it

Linear algebra? Lie understood it much better than we. In thattime some people said Lienear algebra because of him

Quite big calculations (9 cases etc)? Read any paper of Lie andsee how good he was in calculations before asking suchquestions again.

• He did not know the projective invariance of themetrization equation!!!• Message of this lecture: projective invariance isimportant!!!• In the next lecture we will construct another type ofprojectively invariant objects and prove a classicalconjecture with their help

Lecture 4

Plan

Tensor invariants of the projective structure: Weyl and Liouvilletensors

Proof of Lichnerowicz conjecture

What are tensor invariants?

Tensor invariants of a projective structure are tensor fields canonicallyconstructed by an affine connection in the projective structure such thatthey do not depend on the choice of affine connection within thisprojective structure.

Γijk = Γ

ijk + φkδ

ij + φjδ

ik . (∗)

Example. Curvature and Ricci tensors are NOT tensor invariants.Indeed, if we replace a connection Γ by the connection Γ, then the directcalculations using the straightforward formula

Rmikp = ∂kΓ

mip − ∂pΓ

mik + Γa ipΓ

mak − Γa ikΓ

map

give us the following relation between the curvature tensors of Γ and Γ:

Rhijk = Rh

ijk + (φj,k − φk,j)δhi + δhk (φi,j − φiφj)− δhj (φi,k − φiφk) .

Contracting this formula with respect to h, k , we obtain the followingrelation of the Ricci curvatures of Γ and Γ:

Rij = Rij + (n − 1) (φi,j − φiφj) + φi,j − φj,i .

Projective Weyl tensor.

It is the following tensor field:

W hijk = Rh

ijk− 1n−1

(δhkRij − δhjRik

)+ 1

n+1

(

δhiR[jk] − 1n−1

(δhkR[ji ] − δhjR[ki ]

))

.

Theorem (Weyl, Schouten). Projective Weyl tensor is a tensorinvariant of a projective structure, i.e. it does not depend on the choiceof connection within the projective structure.

Proof. Substituting the formulas

Rhijk = R

hijk +

(φj,k − φk,j

)δhi + δ

hk

(φi,j − φiφj

)− δ

hj

(φi,k − φiφk

).

Rij = Rij + (n − 1)(φi,j − φiφj

)+ φi,j − φj,i .

in the definition of the projective Weyl tensor changing the covariantderivative in Γ by the covariant derivative in Γ, we see after an hour ofcalculations that all φ’s disappear.

Liouville invariant

In dimension 2, Weyl tensor is necessary identically zero, since each (1, 3)tensor with its symmetries is zero. Fortunately and exceptionally, there isone more tensor invariant in dimension 2:

Theorem (Liouville 1889). The tensor field Li jk := Ri j,k − Rik,j is atensor invariant in dimension 2.

Proof. Substituting

Rij = Rij + (n − 1)(φi,j − φiφj

)+ φi,j − φj,i .

in the definition of L and changing the covariant derivative in Γ by thecovariant derivative in Γ we again see that all terms containing φdisappear (assuming n = 2).

Remark. There is a similar story in conformal geometry: conformalWeyl tensor vansihes for n ≤ 3 but in dimension 3 there exists anadditional conformal invariant and in dimension 2 conformal geometry isnot interesting all. There is a deep explanation of this similarity and thereare many results in n + 1 dimensional conformal geometry that arevisually similar to results in n-dimensional projective geometry; we willnot discuss in in this lecture course but just remember that many ideasfrom my course can be effectively used in the conformal geometry as well.

How many essential components does Li jk have and whenit vanishes?

Theorem (Liouville 1889). The tensor field Li jk := Ri j,k − Rik,j is a tensor invariant in dim 2.

The tensor Li jk is skew-symmetric in j , k , assuming n = dimM = 2 itimplies that it has two essential components and can be written in theform L = (L1dx

1 + L2dx2)⊗ (dx1 ∧ dx2).

Theorem. Let ∇g = (Γijk) be the Levi-Civita connection of g on 2-dimM. Then, Lijk ≡ 0 if and only if g has constant curvature.

Proof. It is well-known (and follow from the symmetries of the curvaturetensor) that the 2-dim manifold are automatic Einstein in the sense that

Rij =12Rgij .

Calculating Lijk gives

Li jk = Ri j,k − Ri j,k = 12 (R,kgi j − R,jgik) .

Since g is nondegenerate, vanishing of L implies vanishing of dR andhence the constancy of the curvature.

Remark. We also see (or can easily check) that Lijk = dR ⊗ (dx ∧ dy).

W ≡ 0 implies constant curvature

Theorem. Let Γ be the Levi-Civita connection of g on M with n > 2.Then, W h

ijk ≡ 0 if and only if g has constant sectional curvature.

Proof. For Levi-Civita connections the Ricci tensor is symmetric so theformula for W reads

W hijk = Rh

ijk − 1n−1

(δhkRij − δhjRik

).

If W ≡ 0, we obtain

Rhijk = 1

n−1

(δhkRij − δhjRik

).

After lowing the index we have therefore

Rhijk = 1n−1 (ghkRij − ghjRik) .

We see that the left-hand-side is symmetric with respect to(h, i , j , k)←→ (j , k , h, i), so should be the right-hand-side, which impliesthat Rij is proportional to gij , Rij =

Rngij so we have

Rhijk = Rn(n−1) (ghkgij − ghjgik)

which is equivalent to “sectional curvature is constant”.

By-product: Beltrami Theorem

Theorem. Let ∇g = (Γijk ) be the Levi-Civita connection of g on 2-dim M. Then, Lijk ≡ 0 if and only

if g has constant curvature.

Theorem. Let Γ be the Levi-Civita connection of g on M with n > 2. Then, W hijk ≡ 0 if and only if g

has constant sectional curvature.

Corollary (Beltrami Theorem; Beltrami 1865 for dim 2; Schur 1886for dim¿2). A metric projectively equivalent to a metric of constantcurvature has constant curvature.

Nice result for today: projective Lichnerowicz conjecture

Theorem. Let (M, g) be a compact Riemannian manifold such that thesectional curvature is not constant positive. Then, any projective vectorfield is a Killing vector field.

Remark. We have seen in Lecture 3 that the algebra of projective vectorfields of the round sphere is sl(n + 1) and is bigger than the algebra ofisometries which is so(n + 1).

Remark. We have also seen that in dimension 2 there are (local) metricsof nonconstant curvature admitting projective vector fields. One canconstruct similar examples in all dimensions. Theorem above says thatthese examples can not be extended to a closed manifold.

Was a very popular conjecture

Special cases were proved before by French, Japanese and Sovietgeometry schools.

France(Lichnerowicz)

Japan(Yano, Obata, Tanno)

Soviet Union(Raschewskii)

Couty (1961) provedthe conjecture assu-ming that g is Einsteinor Kahler

Yamauchi (1974) pro-ved the conjecture as-suming that the scalarcurvature is constant

Solodovnikov (1956)proved the conjectureassuming that all ob-jects are real analyticand that n ≥ 3.

Remark. Stronger statements are also true:

The statement remains true if one replaces “closed” by“complete”, assumes in addition that the projective vectorfield is complete, and also allows flat metrics:

Theorem. On a compete Riemannian manifold such that itscurvature is not nonnegative constant Proj0 = Iso0.

One can show that on closed manifolds |Proj/Iso| ≤ 2n(Zeghib 2014). Actually, one can even slightly improve theresult:

Theorem (obtained in plane from Munich to Athen). Onclosed manifolds such that the curvature is not positiveconstant |Proj/Iso| ≤ 2

A difficulty of dimensions n ≥ 3 which I avoid by additionalassumption.In dimension 2, in the solution of the 1st Lie problem, we assumedw.l.o.g. that dim(Sol([Γ]) = 2.

The argument was: there exists a 2-dimensional invariant subspace ofSol([Γ] and if the solutions from this subspace are degenerate there existsa Killing vector field.

The latter arguments does not work in dimensions ≥ 3, but still we mayassume that dim(Sol([Γ]) ≤ 2 because of the following nontrivial theoremwhich will not be proved in this lecture. I will posssibly touch it in the 5thlecture.

Theorem (M∼ 2003). On a closed Riemannnian manifold such that itssectional curvature is not constant positive, dim (Sol([Γ])) ≤ 2.

Plan of the proof of the Lichnerowicz conjecture.

Setup.

Our manifold is closed and Riemannian.

The projective structure of the metric admits a projective vectorfield.

We assume that dim (Sol([Γ])) ≤ 2.

Our goal is to show that this vector field is a Killing vector fieldunless g has constant sectional curvature

The case 2 > dim(Sol([Γ]) = 1

If dim (Sol([Γ])) = 1, every two projective related metrics areproportional. Then, a projective vector field v is a homothetyvector field. Since our manifold is closed, every homothety isisometry so our vector field is a Killing vector field as we want.

The case dim(Sol([Γ]) = 2

Important observation already used in the solution of Lieproblems. Lv : Sol([Γ])→ Sol([Γ]), where Lv is the Lie derivative.

After appropriate choice of a basis in Sol([Γ]), we obtained that the v -Lie derivative σ, σ are given by

[Lvσ = λσLv σ = µσ

] [Lvσ = λσ +µσLv σ = −µσ +λσ

] [Lvσ = λσ +σLv σ = λσ

]

.

Thus, the evolution of the solutions along the flow Φt of v is

[Φ∗

t σ = eλtσΦ∗

t σ = eµt σ

]

[Φ∗

t σ = eλt cos(µt)σ +eλt sin(µt)σΦ∗

t σ = −eλt sin(µt)σ +eλt cos(µt)σ

]

[Φ∗

t σ = eλtσ +teλt σΦ∗

t σ = eλt σ

]

.

We will consider all these three cases separately.

The simplest case is when the evolution is given by

[Φ∗

t σ = eλt cos(µt)σ +eλt sin(µt)σΦ∗

t σ = −eλt sin(µt)σ +eλt cos(µt)σ

]

.

Suppose our metric corresponds to the element aσ + bσ.Its evolution is given by

Φ∗t (aσ + bσ) = a(eλt cos(µt)σ + eλt sin(µt)σ)

+b(−eλt sin(µt)σ + eλt cos(µt)σ)

= eλt√a2 + b2(cos(µt + α)σ + sin(µt + α)σ),

where α = arccos(a/(√a2 + b2)).

Now, we use that the metric is Riemannian. Then, for any point x thereexists a basis in TxM such that σ and σ are given by diagonal matrices:σ = diag(s1, s2, ...) and σ = diag(s1, s2, ...).

Then, Φ∗t (aσ + bσ) at this point is also diagonal with the ith element

eλt√a2 + b2(cos(µt + α)si + sin(µt + α)si ).

Clearly, for a certain t we have that Φ∗t (aσ + bσ) is degenerate which

contradicts the assumption,

The proof is is similar when the evolution is given by

[Φ∗

t σ = eλtσ +teλt σΦ∗

t σ = eλt σ

]

.

We again suppose that our metric corresponds to the element aσ + bσ.

Its evolution is given by

Φ∗t (aσ + bσ) = a(eλtσ + eλttσ) + b(eλt σ)

= eλt(aσ + (b + at)σ).

We again see that unless a 6= 0 there exists t such that Φ∗t (aσ + bσ) is

degenerate which contradicts the assumption.

Now, if a = 0, then g corresponds to σ and v is its Killing vector

field,

The most complicated case is when the evolution is given by the matrix

[Φ∗

t σ = eλtσΦ∗

t σ = eµt σ

]

. (2)

The case λ = µ is trivial, in this case the projective vector field ishomothety vector field. We assume λ > µ.

We may assume that g corresponds to the solution σ + σ.Consider, for each t ∈ R, the (1, 1)-tensor

At = (σ + σ)−1Φ∗t (σ + σ) = (σ + σ)−1(eλtσ + eµt σ).

Take a point p and consider a basis such that

g = diag(1, ..., 1), σ = diag(s1, ..., sn), σ = diag(s1, ..., sn)

(Since σ + σ corresponds to g , we have si = 1− si ).

In this basis, we have

At = diag(s1eλt + s1e

µt , ...).

Next, for each t ∈ R, consider the tensor

Gt = g−1Φ∗t g .

Because of the relation g−1 = σ | det(σ)| (see Lecture 2), we have

Gt = diag(

1(s1eλt+s1eµt)

∏

i (si eλt+si eµt)

, ...)

.

Gt = diag(

1(s1eλt+s1eµt)

∏

i (si eλt+si eµt)

, ...)

.

Let us assume for simplicity that all si , si 6= 0. Since λ > µ,

Gtt→+∞−→ diag(e−(n+1)λt , ...) and Gt

t→−∞−→ diag(e(n+1)µt , ...). (⋆)

Consider now the function f = (|W |g )2 = W ijkℓgii ′g

jj′gkk′

g ℓℓ′W i ′

j′k′ℓ′ . Itis a smooth function on the manifold. At points such that W 6= 0 wehave f (p) 6= 0.

Since Φ∗t (g) = gGt and because of (⋆) we have that Φ∗

t (g) hasasymptotic e−2(n+1)λt for t → +∞ and e−2(n+1)µt for t → −∞.Now, because W is projectively invariant, Φ∗

tW = W . Thus,for t →∞,

f (Φt(p)) = Φ∗t f (p) = |Φ∗

tW |2Φ∗

t g∼ const e2(n+1)λt

(where const = 0 iff W = 0)

and for t → −∞ we have f (Φt(p)) ∼ const e−2(n+1)µt .

f (Φt(p)) = Φ∗t f (p) = |Φ∗

tW |2Φ∗

t g∼ const e2(n+1)λt

(where const = 0 iff W = 0) and for t → −∞ we have

f (Φt(p)) ∼ const e−2(n+1)µt

We see that if W (p) 6= 0 then the smooth function f on a compactmanifold is unbounded, which gives a contradiction.

Remark. We had an additional assumption: all si 6= 0. It is not essential,one simply should be slightly more careful.

Remark. In the 2 dim case one should replace W by the Liouvilleinvariant Lijk .

Summary of the proof of the projective Lichnerowiczconjecture

Theorem (Lichnerowicz conjecture). Let (M, g) be a compact Riemannian manifold such that thesectional curvature is not constant positive. Then, any projective vector field is a Killing vector field.

We assumed in addition that dim (Sol([Γ])) = 2 and justified thisassumption by certain fact we did not prove.

Then, we used the invariance of the metrization equation andobtained that the evolution of the solutions along the flow of theprojective vector field is given by one of the three cases:

[Φ∗

t σ = eλtσ

Φ∗

t σ = eµt σ

]

,

[

Φ∗

t σ = eλt cos(µt)σ +eλt sin(µt)σ

Φ∗

t σ = −eλt sin(µt)σ +eλt cos(µt)σ

]

,

[

Φ∗

t σ = eλtσ +teλt σ

Φ∗

t σ = eλt σ

]

.

In all three cases some geometrically constructed (and thereforecontinuous ) function is unbounded which can not happen on a

closed manifold: in the blue and black cases it det(g)det(Φ∗

1 g). In the red

case the function is f = (|W |g )2 = W ijkℓgii ′g

jj′gkk′

g ℓℓ′W i ′

j′k′ℓ′ . It isunbounded unless W ≡ 0. In the proof we have used that W isprojectively invariant, and that W = 0 implies constant curvature.

A bit of philosophy

Felix Klein, 1873, Vergleichende Betrachtungen uberneuere geometrische ForschungenProblem I: Es ist eine Mannigfaltigkeit und in derselben eine Trans-formationsgruppe gegeben. Man entwickle die auf die Gruppe be-zugliche Invariantentheorie.

English Translation. Given a manifoldness and a group oftransformations of the same; to develop the theory of invariants relatingto that group.

In our case we had a manifold, a group of projective transformations, aprojective invariants for them, and used them to prove Lichnerowiczconjecture.

In the previous lecture we also had projectively invariants objects, andthey were extremely effective.

Wait for new projectively-invariant objects in the 5th lecture!

Lecture 5

Plan

Nice result for today (Weyl-Ehlers problem anddim(Sol [Γ]) ≤ 2 for compact manifolds)

Petrov’s solution of the simplest (Riemannian) version of theWeyl-Ehlers problem

Conification and its application: first metric projectively equivalent objects proofs of the results announced above.

Sorry, this time one important statement comes as“black boxe” (=no proof), I will still try to explainthe effects and give the precise references.

Suppose we would like to understand the structure of thespace-time (i.e., a 4-dimensional metric of Lorenz signature) in acertain part of the universe.

We live here

Photo of Pulsar by NASA and ESA

We would like to know what happends here

huge distance

We assume that this part is far enough so the we can use onlytelescopes (in particular we can not send a space ship there).

We still assume that the telescopes can see sufficiently manyobjects in this part of universe.

Then, if the relativistic effects are not negligible (that happens forexample is the objects in this part of space time are sufficiently fastor if this region of the universe is big enough),we obtain as a rule the world lines of the objects asunparameterized curves.

In many cases, we do can get unparameterized geodesicswith the help of astronomic observations

One can obtain unparameterized geodesics by observation:

We take 2 freely falling observers that measure two angular coordinates of the visible objects

Information

and send this information to one place. This place will have 4 functions angle(t) for every visible object which are in the generic case 4 coordinates of the object.

This place has 4=2+2 coordinates of any visiable object

Telsecope N1

Telsecope N2

Information

In many cases, the only thing one can get by observationsare unparametrised geodesics

If one can not register a periodic process on the observed body, one cannot get the own time of the body

T h i s s i t u a t i o n i s e x t r e m e l y r a r e

Problem 1. How to reconstruct a metric by itsunparameterized geodesics?

The mathematical setting: We are given a projective structure givenas in inefficient definition from Lecture 1, as a family γ(t;α) in U ⊆ R

4;we assume that the family is sufficiently big in the sense that ∀x0 ∈ U

Ωx0 := ξ ∈ Tx0U | ∃α and ∃t0 with ddtγ(t;α)|t=t0 is proportional to ξ

contains an open subset of Tx0U.We need to find a metric g such that all γ(t;α) are reparameterizedgeodesics.

The problem was explicitly stated by the famous physists

Jurgen Ehlers 1972, who said that “We reject clocks as ba-sic tools for setting up the space-time geometry and propose... freely falling particles instead. We wish to show how thefull space-time geometry can be synthesized ... . Not only themeasurement of length but also that of time then appears as aderived operation.”

Problem 1 can be naturally divided in two subproblems

Subproblem 1.1. Given a family of curves γ(t; a), how to understandwhether these curves are reparameterised geodesics of a certain affineconnection? How to reconstruct this connection effectively?

In Lecture 1, we considered a two-dimensional analog of this problem,and have seen that 4 geodesics at any points allows us to construct bylinear algebraic manipulations the coefficients of an affine connection inthe projective class.

Now we do the same in any dimension using the same ideas

Subproblem 1.2. Given an affine connection Γ = Γijk , how tounderstand whether there exists a metric g in the projective class of Γ?How to reconstruct this metric effectively?

We know that the existence of the metric is equivalent to the existenceof a nondegenerate solution of the metrization equation; the input of thislecture is few tricks that help.

Problem 2 (implicitely, Weyl). In what situationsinteresting for physics the reconstruction of a metric by theunparameterised geodesics is unique (up to themultiplication of the metric by a constant)?

In other words, what metrics ’interesting’ for relativity allow nontrivialprojective equivalence?

We already have seen (Lagrange example) that constant curvaturemetrics have many projectively equivalent metrics (all having constantcurvature). Let us construct one more example interesting for physics.

Example. The so-called Friedman-Lemaitre-Robertson-Walker metric

g = −dt2 + R(t)2dx2 + dy2 + dz2

1 + κ4 (x

2 + y2 + z2); κ = +1; 0;−1,

is not projectively rigid.

Indeed, ∀c the metric

g =−1

(R(t)2 + c)2dt2 +

R(t)2

c(R(t)2 + c)

dx2 + dy2 + dz2

1 + κ4 (x

2 + y2 + z2)

is geodesically equivalent to g (essentially Levi-Civita 1896; repeated bymany relativists (Nurowski, Gibbons et al, Hall) later).

One can of course check that the metrics are projectively equivalent bydirect calculations.

Goals: physically interesting projectively rigid examples

Theorem (Petrov 1961). Let g and g be two projectively equivalentRicci-flat 4 dim metrics (of arbitrary siganture). Then, they are flat orproportional.

I will give a proof of this results in the Riemannian case, this will be easyand requires no theory. All other results will need some additional resultswhich I will introduce as black boxes (=without proofs).

Theorem (Kiosak-M∼ 2009). Let g and g be projectively equivalent4 dim metrics of arbitrary signature. Assume g is Einstein (i.e.,Ricc = Scal

4 g). Then, Levi-Civita connections of g and g coincide unlessmetrics have constant curvature.

There exist counterexamples in higher dimensions. By the next theorem,counterexamples are local.

Theorem (Kiosak-M∼ 2012 + Mounoud-M∼ 2013). Let g and gbe projectively equivalent metrics of arbitrary signature on a compactmanifold of dimension ≥ 2. Assume g is Einstein. Then, Levi-Civitaconnections of g and g coincide unless metrics have constant curvature.

I will also explain the statement we have used in the proofof the Lichnerowciz conjecture

Theorem (Kiosak - M∼ 2012+ M∼-Mounoud 2013)(Riemannian case: M∼ 2003). On a closed manifold of arbitrarycurvature such that its sectional curvature is not constant,dim (Sol([Γ])) ≤ 2.

Algorithm how to reconstruct Γ by sufficiently manygeodesics

Repeat: d2γa

dt2+ Γabc

dγb

dtdγc

dt= f

(dγdt

)dγa

dt. (∗)

Take a point x0; our goal it to reconstruct [Γ(x0)ijk ]. Take γ(t0;α) such

that γ(t0;α) = x0 and the first component(

dγ1

dt

)

|t=t06= 0. For γ(t0;α),

we rewrite the equation (∗) at t = t0 in the following form:

f(

dγdt

)

=(d2γ1

d2t+ Γ1ab

dγa

dtdγb

dt

)/ dγ1

dt

dγ2

dtΓ1ab

dγa

dtdγb

dt− dγ1

dtΓ2ab

dγa

dtdγb

dt= d2γ2

d2tdγ1

dt− dγ2

dtd2γ1

d2t...

dγn

dtΓ1ab

dγa

dtdγb

dt− dγ1

dtΓnab

dγa

dtdγb

dt= d2γn

d2tdγ1

dt− dγn

dtd2γ1

d2t.

(3)

The first equation of (3) is equivalent to the equation (∗) for a = 1

solved with respect to f(

dγdt

)

. We obtain the second, third, etc,

equations of (3) by substituting the first equation of (3) in the equations(∗) with a = 2, 3, etc.

Note that the subsystem of (3) containing the the second, third, etc.equations of (3) does not contain the function f and is therefore a linearsystem on Γijk .

dγ2

dtΓ1ab

dγa

dtdγb

dt− dγ1

dtΓ2ab

dγa

dtdγb

dt= d2γ2

d2tdγ1

dt− dγ2

dtd2γ1

d2t...

dγn

dtΓ1ab

dγa

dtdγb

dt− dγ1

dtΓnab

dγa

dtdγb

dt= d2γn

d2tdγ1

dt− dγn

dtd2γ1

d2t.

(3′)

Then, for every ‘geodesic’ γ(t0, α) gives us n− 1 linear (inhomogeneous)equations on the components Γ(x0)

ijk . We take a sufficiently big number

N (if n = 4, it is sufficient to take N = 12) and substitute N genericgeodesics γ(t;α) passing through x0 in this subsystem.

At every point x0, we obtain an inhomogeneous linear system of

equations on n2(n+1)2 unknowns Γ(x0)

ijk .

In the case the solution of this system does not exist (at least at one

point x0), there exists no connection whose (reparameterized) geodesics

are γ(t;α). In the case it exists, the solution of the system above gives

us the projective class of the connection.

Proof of Petrov’s result for Riemannian metrics

Theorem (Petrov 1961). Let g and g be two projectively equivalentRiemannian 4 dim Ricci-flat metrics. Then, they are flat or proportional.

Proof. Consider the projective Weyl tensor

Wijkℓ := R

ijkℓ − 1

n−1

(

δiℓ Rjk − δ

ik Rjℓ

)

We know (Lecture 4) the the projective Weyl tensor does not depend ofthe choice of metric within the projective class.Now, from the formula for the Weyl tensor, we know that, if thesearched g is Ricci-flat, projective Weyl tensor coincides with theRiemann tensor R i

jkℓ of g . Thus, if we know the projective class of theRicci-flat metric g , we know its Riemann tensor.

Then, the metric g must satisfy the following system of equations due tothe symmetries of the Riemann tensor:

giaW

ajkm + gjaW

aikm = 0

giaWajkm − gkaW

amij = 0

(4)

The first portion of the equations is due to the symmetry(Rijkm = −Rjikm), and the second portion is due to the symmetry(Rkmij = Rijkm) of the curvature tensor of g .

We see that for every point x0 ∈ U the system (4) is a system of linearequations on g(x0)ij . The number of equations (around 100) is muchbigger than the number of unknowns (which is 10). It is expectedtherefore, that a generic projective Weyl tensor W i

jkl admits no morethan one-dimensional space of solutions (by assumtions, our W admits atleast one-dimensional space of solutions). The expectation is true, as thefollowing classical result shows

Theorem (Folklore – Petrov, Hall, Rendall, Mcintosh) Let W ijkℓ

be a tensor in R4 such that it is skew-symmetric with respect to k , ℓ and

such that its traces W aakℓ and W a

jaℓ vanish. Then, if W 6= 0, twopositively definite solutions of the equations (4) are proportional.

By this theorem, the metrics g and g are conformally equivalent which

by result of Weyl implies that they are proportional

The trick in proof of Petrov’s result simplifies thereconstruction of the metric

We have seen that (in dim 4) given [Γ] which is not flat we can constructW and in the Riemannian case or under additional mild assumptions onW the conformal class of the metric g .

Then, the all (there are 20 of them) metrization equations are equationson ONE unknown function and one can be solved by integration of anexplicitly given 1-form (M∼-Trautmann 2014).

Main tool in the proof of all others theorem: conification

Let M be a manifold and g =

gij

a Riemannian metric on

it. The cone over this manifold is a manifold R>0 ×M with the

metric dt2 + t2g =

1

t2gij

Geometric picture behindthe word “conification”

O

(M,g)

Think that the manifold Mn (orarbitrary dimension n) is imbed-ded in the sphere (of arbitrarydimension N ≥ n) and carriesthe induced metric g . Considerthe union of all rays connectingthe origin of the sphere with thepoints of M. Then, this unionis a n + 1-dimensional manifold,and the restriction of the standardEuclidean metric to it is the conemetric.

Metrization equations in the presence of metric

Recall that the solutions σij of the metrization equations are weightedtensors.In the case we work with the Levi-Civita connection of a metric, we canchoose Volg as the reference volume form. Then, weighted tensors can beviewed as tensors. Since the volume form is parallel, the covariantderivatives is the usual covariant derviative of tensor fields, andmetrization equations can be rewritten as below:

Theorem (Sinjukov 1962). Let g be a metric. The metrics g that areprojectively equivalent to g are in one-to-one correspondence with thesolutions of the following system of PDE on the (0,2)-tensorfield a = aijand (0,1)-tensorfield λi such that det(a) 6= 0 at all points:

aij,k = λigjk + λjgik . (∗)The one-to-one correspondence is given by

g −→

a =

(det(g)

det(g)

) 1n+1

gg−1g , λ = 12dtraceg (a)

.

Black-Box statement

Thm (Kiosak-M∼ 2011/2013). Let g be a metric on ann ≥ 3-dimensional connected manifold such that dim Sol ≥ 3 or g isEinstein.Then, there exists a constant B such that for any solution (a, λ) of theequations aij,k = λigjk + λjgik there exists a function µ such that inaddition the following two equations are fulfilled:

λi,j = µgij − Baij

µ,i = 2Bλi .

In the case g is Einstein B = Scaln(n−1)

Remark. Proof is technically nontrivial, but standard: We went until5th prolongation in the Cartan-Kahler prolongation procedure to prove it.

Remark. The constant B depends on the metric but is the same for allsolutions (a, λ). We assume B 6= 0, in this case one can make it 1 byscaling the metric.

Projectively equivalent metrics as parallel (0,2)-tensors onthe cone.

What is explained on the previous slide. We assume that n =

dim(M) ≥ 3 and dim Sol ≥ 3 or g is Einstein. Then, metrics g

that are projectively equivalent to g are in one-to-one correspondence

to the solutions (a, λ, µ) of the following system of equations

aij,k = λigjk + λjgik

λi,j = µgij − aij

µ,i = 2λi

Principal Observation. The solutions of these equations are inone-to-one correspondence with the parallel symmetric (0, 2)-tensors onthe cone (M, g) = (R>0 ×M, dt2 + t2g).

The one-to-one correspondence is given by

(a, λ, µ) 7→ A :=

µ −t · λ1 . . . −t · λn

−t · λ1 t2 · a11 . . . t2 · a1n...

......

−t · λn t2 · an1 . . . t2 · ann

.

Proof is an easy exercise – write down the Levi-Civita connection ofthe cone metric dt2 + t2g and see that the condition that A is parallel isequivalent to the above equations on a, λ, µ.I do not have any geometric explanation of this phenomenon.

Principal Observation. Under our assumptions, metrics projec-tively equivalent to g are essentailly the same as parallel symmetric(0,2)-tensors on (M, g) = (R>0 ×M, dt2 + t2g).

Corollary. Levi-Civita connection of (M, g) does not depend on thechoice of the metric g in the projective class.

This is metrically projecively invariant object! In order to constructit, we need the metric, but it does not depend on the choice of themetric in the projective class.Corollary. The projection to M of the Riemannian curvature and of theRicci tensor to the manifold is projectively invariant.

On the manifold M, these projections looks as follows:

Z ijkℓ := R i

jkℓ +(δi ℓgjk − δi kgjℓ

)= R − K ,

where K is the ”algebraic constant curvature tensor”; and the projectionof Ricc is its trace. For Einstein metrics, this gives nothing new, since inthe Einstein case Z is the projective Weyl tensor, and its trace vanishes.But in the case dim Sol ≥ 3 is does give new invariants.

These are again metric projective invariants!

Cones over compact manifolds do not have parallel(0,2)-tensors

Theorem ( Tanno 1978, Obata 1978, M∼- Mounoud 2013).Non flat cones over closed manifolds do not admit parallelsymmetric (0, 2) tensors nonproportional to the metrics.

This Theorem proves all announced above Theorems.Proof in the Riemannian case. The existence of nontrivialparallel tensor implies the existence of the local decompositiong = g1 + g2.

Since cone merics admit homotheties, the metrics g1 and g2 admithomotheties. One can show the existence of a stable point whichby blow up argument of Gromov implies that they are flat